# Seminar at SMU Delhi

February 18, 2014 (Tuesday) , 3:30 PM at Webinar
Speaker: Yuri Bilu, University of Bordeaux, France
Title: Drawing curves on checkered paper.
Abstract of Talk
Imagine a sheet of checkered paper, the one used in arithmetic classes in primary schools. Let us try to draw a curve on this sheet which would intersect as many crossings'' as possible. (In other words: how many lattice points a compact curve can meet?) Of course, one can draw a rather curvy'' curve which would pass through every crossing. But the problem becomes interesting if the curve is assumed not too curvy''. For instance, in 1927 the Czech mathematician Jarnik proved that a *strictly convex* curve can pass at most $O(N^{2/3})$ crossings on an $N \times N$ checkered sheet. I will prove the theorem of Jarnik using an argument suggested by the German mathematician Dörge (1926), who worked independently of Jarnik on related problems. I will then show how a slight modification of D\"orge's argument leads to a wonderful theorem of Bombieri and Pila (1989) on counting lattice points on analytic curves. If time permits, I will say a few words how in the last years these ideas were successfully used by Zannier, Pila and others to attack some prominent conjectures.